In mathematics, a greatest common divisor matrix (sometimes abbreviated as GCD matrix) is a matrix.

Definition

111111111 1
121212121 2
113113113 1
121412141 2
111151111 5
123216123 2
111111711 1
121412181 2
113113119 1
12125212110
GCD matrix of (1,2,3,...,10)

Let be a list of positive integers. Then the matrix having the greatest common divisor as its entry is referred to as the GCD matrix on .The LCM matrix is defined analogously.[1][2]

The study of GCD type matrices originates from Smith (1875) who evaluated the determinant of certain GCD and LCM matrices. Smith showed among others that the determinant of the matrix is , where is Euler's totient function.[3]

Bourque–Ligh conjecture

Bourque & Ligh (1992) conjectured that the LCM matrix on a GCD-closed set is nonsingular.[1] This conjecture was shown to be false by Haukkanen, Wang & Sillanpää (1997) and subsequently by Hong (1999).[4][2] A lattice-theoretic approach is provided by Korkee, Mattila & Haukkanen (2019).[5]

References

  1. 1 2 Bourque, K.; Ligh, S. (1992). "On GCD and LCM matrices". Linear Algebra and Its Applications. 174: 65–74. doi:10.1016/0024-3795(92)90042-9.
  2. 1 2 Hong, S. (1999). "On the Bourque–Ligh conjecture of least common multiple matrices". Journal of Algebra. 218: 216–228. doi:10.1006/jabr.1998.7844.
  3. Smith, H. J. S. (1875). "On the value of a certain arithmetical determinant". Proceedings of the London Mathematical Society. 1: 208–213. doi:10.1112/plms/s1-7.1.208.
  4. Haukkanen, P.; Wang, J.; Sillanpää, J. (1997). "On Smith's determinant". Linear Algebra and Its Applications. 258: 251–269. doi:10.1016/S0024-3795(96)00192-9.
  5. Korkee, I.; Mattila, M.; Haukkanen, P. (2019). "A lattice-theoretic approach to the Bourque–Ligh conjecture". Linear and Multilinear Algebra. 67 (12): 2471–2487. arXiv:1403.5428. doi:10.1080/03081087.2018.1494695. S2CID 117112282.
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